It Doesn't Always Work: The Pitfall of Judging Skewness by the Mean < Median < Mode Rule

     You’ve probably heard — or even taught — the classic rule:

·    If the mean is less than the median, the distribution is skewed to the left;

·    If the mean is greater than the median, the skewness is to the right;

·    If mean, median, and mode coincide, the distribution is symmetric.

This rule works in many situations... but it’s not universal. Relying on it too blindly can lead to serious mistakes, even with small, simple, unimodal distributions.

 📊 A numerical example

Consider this dataset:

0; 0; 0; 0.5; 0.5; 1.05; 1.25

Let’s break it down:

·        Mode: 0

·        Median: 0.5

·        Mean: approximately 0.471

The mean is less than the median, which would suggest left skewness.

But when we compute the skewness coefficient (based on the 3rd moment):

·    Skewness ≈ +0.59, meaning positive skewness → right-skewed distribution!

 🔹 So what’s going on?

The mean–median–mode rule only holds under ideal conditions, such as:

·        Continuous distributions;

·        Smooth shapes (no jumps or plateaus);

·        Well-behaved unimodal structure;

·        No outliers that pull the mean disproportionately.

Our example is small and discrete. The mode has little “weight,” and two larger values pull the mean upward, producing positive skewness, even though the mean is less than the median.

 🤔 A student's question

A student once asked:

“Can the mean be less than the median, and yet the skewness be positive?”

The answer is: yes. And not only is it possible — it’s quite common in small samples or discrete distributions.

 📅 The takeaway

Don’t use the position of the mean alone to assess skewness.

Statistical skewness reflects deeper aspects of the distribution — such as central moments or the overall shape of the histogram.

🔹 It's better to compute the skewness coefficient than to rely blindly on a rule of thumb.

 

Reference

1.                von HIPPEL, P. T. Mean, median and skew: correcting a textbook rule. The Ohio State University.  Journal od Statistics Education. 13(2): 2005.